Inference Rules

From Event-B
Revision as of 15:46, 27 May 2013 by imported>Josselin (Fix rules HYP and CNTR (case distinction on relational predicate))
Jump to navigationJump to search 
CAUTION! Any modification to this page shall be announced on the User mailing list!

Rules that are marked with a * in the first column are implemented in the latest version of Rodin. Rules without a * are planned to be implemented in future versions. Other conventions used in these tables are described in The_Proving_Perspective_(Rodin_User_Manual)#Inference_Rules.


  Name Rule Side Condition A/M


*
HYP
 \frac{}{\textbf{H},\textbf{P}^{\dagger} \;\;\vdash \;\; \textbf{P}} see below for \textbf{P}^{\dagger} A


*
HYP_OR
 \frac{}{\textbf{H},\textbf{Q} \;\;\vdash \;\; \textbf{P} \lor \ldots \lor \textbf{Q} \lor \ldots \lor \textbf{R}} A


*
CNTR
 \frac{}{\textbf{H},\;\textbf{P},\;\textbf{nP}^{\dagger} \;\;\vdash \;\; \textbf{Q}} A


*
FALSE_HYP
 \frac{}{\textbf{H},\bfalse \;\;\vdash \;\; \textbf{P}} A


*
TRUE_GOAL
 \frac{}{\textbf{H} \;\;\vdash \;\; \btrue} A


*
FUN_GOAL
 \frac{}{\textbf{H},\; f\in E\;\mathit{op}\;F \;\;\vdash\;\; f\in T_1\pfun T_2} where T_1 and T_2 denote types and \mathit{op} is one of \pfun, \tfun, \pinj, \tinj, \psur, \tsur, \tbij. A


*
FUN_IMAGE_GOAL
 \frac{\textbf{H},\; f\in S_1\;\mathit{op}\;S_2,\; f(E)\in S_2\;\;\vdash\;\; \mathbf{P}(f(E))}{\textbf{H},\; f\in S_1\;\mathit{op}\;S_2\;\;\vdash\;\; \mathbf{P}(f(E))} where \mathit{op} denotes a set of relations (any arrow) and \mathbf{P} is WD strict M


FUN_GOAL_REC
 \frac{}{\textbf{H},\; f\in S_1\;\mathit{op_1}\;(S_2\;\mathit{op_2}\;(\ldots(S_n\;\mathit{op_n}(U\;\mathit{opf}\;V\;))\ldots)) \;\vdash\;\; f(E_1)(E_2)...(E_n)\in T_1\pfun T_2} where T_1 and T_2 denote types, \mathit{op} denotes a set of relations (any arrow) and \mathit{opf} is one of \pfun, \tfun, \pinj, \tinj, \psur, \tsur, \tbij. A


*
DBL_HYP
 \frac{\textbf{H},\;\textbf{P} \;\;\vdash \;\; \textbf{Q}}{\textbf{H},\;\textbf{P},\;\textbf{P} \;\;\vdash \;\; \textbf{Q}} A


*
AND_L
 \frac{\textbf{H},\textbf{P},\textbf{Q} \; \; \vdash \; \; \textbf{R}}{\textbf{H},\; \textbf{P} \land \textbf{Q} \; \; \vdash \; \; \textbf{R}} A


*
AND_R
 \frac{\textbf{H} \; \; \vdash \; \; \textbf{P} \qquad \textbf{H} \; \; \vdash \; \; \textbf{Q}}{\textbf{H} \; \; \vdash \; \; \textbf{P} \; \land \; \textbf{Q}} A


IMP_L1
 \frac{\textbf{H},\; \textbf{Q},\; \textbf{P} \land \ldots \land \textbf{R} \limp \textbf{S} \;\;\vdash \;\; \textbf{T}}{\textbf{H},\; \textbf{Q},\; \textbf{P} \land \ldots \land \textbf{Q} \land \ldots \land \textbf{R} \limp \textbf{S} \;\;\vdash \;\; \textbf{T} } A


*
IMP_R
 \frac{\textbf{H}, \textbf{P} \;\;\vdash \;\; \textbf{Q}}{\textbf{H} \;\;\vdash \;\; \textbf{P} \limp \textbf{Q}} A


*
IMP_AND_L
 \frac{\textbf{H},\textbf{P} \limp \textbf{Q}, \textbf{P} \limp \textbf{R}\;\;\vdash \;\; \textbf{S}}{\textbf{H},\;\textbf{P} \limp \textbf{Q} \land \textbf{R} \;\;\vdash \;\; \textbf{S}} A


*
IMP_OR_L
 \frac{ \textbf{H},\textbf{P} \limp \textbf{R}, \textbf{Q} \limp \textbf{R}\;\;\vdash \;\; \textbf{S} }{\textbf{H},\;\textbf{P} \lor \textbf{Q} \limp \textbf{R} \;\;\vdash \;\; \textbf{S}} A


*
AUTO_MH
 \frac{ \textbf{H},\textbf{P},\;\textbf{Q}\limp \textbf{R}\;\;\vdash \;\; \textbf{S} }{\textbf{H},\;\textbf{P},\; \textbf{P} \land \textbf{Q} \limp \textbf{R} \;\;\vdash \;\; \textbf{S}} A


*
NEG_IN_L
 \frac{\textbf{H},\; E \in \{ a,\ldots , c\},\; \neg\, (E=b) \; \; \vdash \; \; \textbf{P} }{\textbf{H},\; E \in \{ a,\ldots , b, \ldots , c\} , \neg \, (E=b) \; \; \vdash \; \; \textbf{P} } A


*
NEG_IN_R
 \frac{\textbf{H},\; E \in \{ a,\ldots , c\},\; \neg\, (b=E) \; \; \vdash \; \; \textbf{P} }{\textbf{H},\; E \in \{ a,\ldots , b, \ldots , c\} , \neg \, (b=E) \; \; \vdash \; \; \textbf{P} } A


*
XST_L
 \frac{\textbf{H},\; \textbf{P(x)} \; \; \vdash \; \; \textbf{Q} }{ \textbf{H},\; \exists \, \textbf{x}\, \qdot\, \textbf{P(x)} \; \; \vdash \; \; \textbf{Q} } A


*
ALL_R
 \frac{\textbf{H}\; \; \vdash \; \; \textbf{P(x)} }{ \textbf{H} \; \; \vdash \; \; \forall \textbf{x}\, \qdot\, \textbf{P(x)} } A


*
EQL_LR
 \frac{\textbf{H(E)} \; \; \vdash \; \; \textbf{P(E)} }{\textbf{H(x)},\; x=E \; \; \vdash \; \; \textbf{P(x)} } x is a variable which is not free in E A


*
EQL_RL
 \frac{\textbf{H(E)} \; \; \vdash \; \; \textbf{P(E)} }{\textbf{H(x)},\; E=x \; \; \vdash \; \; \textbf{P(x)} } x is a variable which is not free in E A


SUBSET_INTER
 \frac{\textbf{H},\;\textbf{T} \subseteq \textbf{U} \;\;\vdash \;\; \textbf{G}(\textbf{S} \binter \dots \binter \textbf{T} \binter \dots \binter \textbf{V})} {\textbf{H},\;\textbf{T} \subseteq \textbf{U} \;\;\vdash \;\; \textbf{G}(\textbf{S} \binter \dots \binter \textbf{T} \binter \dots \binter \textbf{U} \binter \dots \binter \textbf{V})} where \mathbf{T} and \mathbf{U} are not bound by \mathbf{G} A


IN_INTER
 \frac{\textbf{H},\;\textbf{E} \in \textbf{T} \;\;\vdash \;\; \textbf{G}(\textbf{S} \binter \dots \binter \{\textbf{E}\} \binter \dots \binter \textbf{U})} {\textbf{H},\;\textbf{E} \in \textbf{T} \;\;\vdash \;\; \textbf{G}(\textbf{S} \binter \dots \binter \{\textbf{E}\} \binter \dots \binter \textbf{T} \binter \dots \binter \textbf{U})} where \mathbf{E} and \mathbf{T} are not bound by \mathbf{G} A


NOTIN_INTER
 \frac{\textbf{H},\;\lnot\;\textbf{E} \in \textbf{T} \;\;\vdash \;\; \textbf{G}(\emptyset)} {\textbf{H},\;\lnot\;\textbf{E} \in \textbf{T} \;\;\vdash \;\; \textbf{G}(\textbf{S} \binter \dots \binter \{\textbf{E}\} \binter \dots \binter \textbf{T} \binter \dots \binter \textbf{U})} where \mathbf{E} and \mathbf{T} are not bound by \mathbf{G} A


*
FIN_L_LOWER_BOUND_L
 \frac{}{\textbf{H},\;\finite(S) \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; n \leq x)} The goal is discharged A


*
FIN_L_LOWER_BOUND_R
 \frac{}{\textbf{H},\;\finite(S) \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; x \geq n)} The goal is discharged A


*
FIN_L_UPPER_BOUND_L
 \frac{}{\textbf{H},\;\finite(S) \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; n \geq x)} The goal is discharged A


*
FIN_L_UPPER_BOUND_R
 \frac{}{\textbf{H},\;\finite(S) \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; x \leq n)} The goal is discharged A


*
CONTRADICT_L
 \frac{\textbf{H},\;\neg\,\textbf{Q} \;\;\vdash \;\; \neg\,\textbf{P}}{\textbf{H},\;\textbf{P} \;\;\vdash \;\; \textbf{Q}} M


*
CONTRADICT_R
 \frac{\textbf{H},\;\neg\,\textbf{Q} \;\;\vdash \;\; \bfalse}{\textbf{H} \;\;\vdash \;\; \textbf{Q}} M


*
CASE
 \frac{\textbf{H}, \; \textbf{P} \; \; \vdash \; \; \textbf{R} \qquad\ldots\qquad \textbf{H}, \; \textbf{Q} \; \; \vdash \; \; \textbf{R} }{\textbf{H},\; \textbf{P} \lor \ldots \lor \textbf{Q} \; \; \vdash \; \; \textbf{R} } M


*
IMP_CASE
 \frac{\textbf{H}, \; \lnot\textbf{P} \; \; \vdash \; \; \textbf{R} \qquad \textbf{H}, \; \textbf{Q} \; \; \vdash \; \; \textbf{R} }{\textbf{H},\; \textbf{P} \limp\textbf{Q} \; \; \vdash \; \; \textbf{R} } M


*
MH
 \frac{\textbf{H} \;\;\vdash\;\;\textbf{P} \qquad \textbf{H},\; \textbf{Q} \;\;\vdash \;\; \textbf{R} }{\textbf{H},\;\textbf{P} \limp \textbf{Q} \;\;\vdash \;\; \textbf{R}} M


*
HM
 \frac{\textbf{H} \;\;\vdash\;\;\neg\,\textbf{Q} \qquad \textbf{H},\; \neg\,\textbf{P} \;\;\vdash \;\; \textbf{R} }{\textbf{H},\;\textbf{P} \limp \textbf{Q} \;\;\vdash \;\; \textbf{R}} M


EQV
 \frac{\textbf{H(Q)},\; \textbf{P} \leqv \textbf{Q} \;\;\vdash\;\; \textbf{G(Q)}}{\textbf{H(P)},\;\textbf{P} \leqv \textbf{Q} \;\;\vdash \;\; \textbf{G(P)}} M


*
OV_SETENUM_L
 \frac{\textbf{H},\; G=E ,\;\textbf{P}(F)\;\;\vdash\;\;\textbf{Q} \qquad \textbf{H},\; \neg\,(G=E) ,\;\textbf{P}((\{E\}) \domsub f)(G))\;\;\vdash\;\;\textbf{Q}}{\textbf{H},\;\textbf{P}((f\ovl\{E \mapsto F\})(G)) \;\;\vdash \;\; \textbf{Q}} where \mathbf{P} is WD strict A


*
OV_SETENUM_R
 \frac{\textbf{H},\; G=E \;\;\vdash\;\;\textbf{Q}(F) \qquad \textbf{H},\; \neg\,(G=E) \;\;\vdash\;\;\textbf{Q}((\{E\}) \domsub f)(G))}{\textbf{H} \;\;\vdash \;\; \textbf{Q}((f\ovl\{E \mapsto F\})(G))} where \mathbf{Q} is WD strict A


*
OV_L
 \frac{\textbf{H},\; G \in \dom(g) ,\;\textbf{P}(g(G))\;\;\vdash\;\;\textbf{Q} \qquad \textbf{H},\; \neg\,G \in \dom(g) ,\;\textbf{P}((\dom(g) \domsub f)(G))\;\;\vdash\;\;\textbf{Q}}{\textbf{H},\;\textbf{P}((f\ovl g)(G)) \;\;\vdash \;\; \textbf{Q}} where \mathbf{P} is WD strict A


*
OV_R
 \frac{\textbf{H},\; G \in \dom(g) \;\;\vdash\;\;\textbf{Q}(g(G)) \qquad \textbf{H},\; \neg\, G \in \dom(g) \;\;\vdash\;\;\textbf{Q}((\dom(g) \domsub f)(G))}{\textbf{H} \;\;\vdash \;\; \textbf{Q}((f\ovl g)(G))} where \mathbf{Q} is WD strict A


*
DIS_BINTER_R
 \frac{\textbf{H} \;\;\vdash\;\; f^{-1} \in A \pfun B \qquad\textbf{H} \;\;\vdash\;\;\textbf{Q}(f[S] \binter f[T]) }{\textbf{H} \;\;\vdash \;\; \textbf{Q}(f[S \binter T])} where A and B denote types. M


*
DIS_BINTER_L
 \frac{\textbf{H} \;\;\vdash\;\; f^{-1} \in A \pfun B \qquad\textbf{H},\;\textbf{Q}(f[S] \binter f[T]) \;\;\vdash\;\;\textbf{G}}{\textbf{H},\; \textbf{Q}(f[S \binter T]) \;\;\vdash \;\; \textbf{G}} where A and B denote types. M


*
DIS_SETMINUS_R
 \frac{\textbf{H} \;\;\vdash\;\; f^{-1} \in A \pfun B \qquad\textbf{H} \;\;\vdash\;\;\textbf{Q}(f[S] \setminus f[T]) }{\textbf{H} \;\;\vdash \;\; \textbf{Q}(f[S \setminus T])} where A and B denote types. M


*
DIS_SETMINUS_L
 \frac{\textbf{H} \;\;\vdash\;\; f^{-1} \in A \pfun B \qquad\textbf{H},\;\textbf{Q}(f[S] \setminus f[T]) \;\;\vdash\;\; \textbf{G}}{\textbf{H},\; \textbf{Q}(f[S \setminus T]) \;\;\vdash \;\; \textbf{G}} where A and B denote types. M


*
SIM_REL_IMAGE_R
 \frac{\textbf{H} \; \; \vdash \; \; {WD}(\textbf{Q}(\{ f(E)\} )) \qquad\textbf{H} \; \; \vdash \; \; \textbf{Q}(\{ f(E)\} ) }{\textbf{H} \; \; \vdash \; \; \textbf{Q}(f[\{ E\} ])} M


*
SIM_REL_IMAGE_L
 \frac{\textbf{H} \; \; \vdash \; \; {WD}(\textbf{Q}(\{ f(E)\} )) \qquad\textbf{H},\; \textbf{Q}(\{ f(E)\}) \;\;\vdash\;\; \textbf{G}}{\textbf{H},\; \textbf{Q}(f[\{ E\} ]) \;\;\vdash\;\; \textbf{G} } M


*
SIM_FCOMP_R
 \frac{\textbf{H} \;\;\vdash\;\;{WD}(\textbf{Q}(g(f(x)))) \qquad\textbf{H} \;\;\vdash\;\;\textbf{Q}(g(f(x))) }{\textbf{H} \;\;\vdash \;\; \textbf{Q}((f \fcomp g)(x))} M


*
SIM_FCOMP_L
 \frac{\textbf{H} \;\;\vdash\;\;{WD}(\textbf{Q}(g(f(x)))) \qquad\textbf{H},\; \textbf{Q}(g(f(x))) \;\;\vdash\;\; \textbf{G}}{\textbf{H},\; \textbf{Q}((f \fcomp g)(x)) \;\;\vdash \;\; \textbf{G}} M


*
FIN_SUBSETEQ_R
 \frac{\textbf{H} \;\;\vdash\;\;{WD}(T) \qquad\textbf{H} \;\;\vdash \;\; S \subseteq T \qquad \textbf{H} \;\;\vdash \;\; \finite\,(T)}{\textbf{H} \;\;\vdash \;\; \finite\,(S)} the user has to write the set corresponding to T in the editing area of the Proof Control Window M


*
FIN_BINTER_R
 \frac{\textbf{H} \;\;\vdash \;\;\finite\,(S) \;\lor\;\ldots \;\lor\; \finite\,(T)}{\textbf{H} \;\;\vdash \;\; \finite\,(S \;\binter\;\ldots \;\binter\; T)} M


FIN_KINTER_R
 \frac{\textbf{H} \;\;\vdash \;\;\exists s\, \qdot\, s \in S \land \finite\,(s)}{\textbf{H} \;\;\vdash \;\; \finite\,(\inter(S))} where s is fresh M


FIN_QINTER_R
 \frac{\textbf{H} \;\;\vdash \;\;\exists s\, \qdot\, P \land \finite\,(E)}{\textbf{H} \;\;\vdash \;\; \finite\,(\Inter s\,\qdot\,P\,\mid\,E)} M


*
FIN_SETMINUS_R
 \frac{\textbf{H} \;\;\vdash \;\;\finite\,(S)}{\textbf{H} \;\;\vdash \;\; \finite\,(S \;\setminus\; T)} M


FIN_REL
 \frac{}{\textbf{H},\; r\in S\;\mathit{op}\;T,\; \finite\,(S),\; \finite\,(T) \;\;\vdash \;\; \finite\,(r)} where \mathit{op} denotes a set of relations (any arrow) A


*
FIN_REL_R
 \frac{\textbf{H} \;\;\vdash\;\;{WD}(S\rel T) \qquad\textbf{H} \;\;\vdash \;\; r \;\in\; S \rel T \qquad \textbf{H} \;\;\vdash \;\; \finite\,(S) \qquad \textbf{H} \;\;\vdash \;\; \finite\,(T)}{\textbf{H} \;\;\vdash \;\; \finite\,(r)} the user has to write the set corresponding to S \rel T in the editing area of the Proof Control Window M


*
FIN_REL_IMG_R
 \frac{\textbf{H} \;\;\vdash \;\; \finite\,(r) }{\textbf{H} \;\;\vdash \;\; \finite\,(r[s])} M


*
FIN_REL_RAN_R
 \frac{\textbf{H} \;\;\vdash \;\; \finite\,(r) }{\textbf{H} \;\;\vdash \;\; \finite\,(\ran(r))} M


*
FIN_REL_DOM_R
 \frac{\textbf{H} \;\;\vdash \;\; \finite\,(r) }{\textbf{H} \;\;\vdash \;\; \finite\,(\dom(r))} M


FIN_FUN_DOM
 \frac{}{\textbf{H},\; f\in S\;\mathit{op}\;T,\; \finite\,(S) \;\;\vdash \;\; \finite\,(f)} where \mathit{op} is one of \pfun, \tfun, \pinj, \tinj, \psur, \tsur, \tbij A


FIN_FUN_RAN
 \frac{}{\textbf{H},\; f\in S\;\mathit{op}\;T,\; \finite\,(T) \;\;\vdash \;\; \finite\,(f)} where \mathit{op} is one of \pinj, \tinj, \tbij A


*
FIN_FUN1_R
 \frac{\textbf{H} \;\;\vdash\;\;{WD}(S\pfun T) \qquad\textbf{H} \;\;\vdash \;\; f \;\in\; S \pfun T \qquad \textbf{H} \;\;\vdash \;\; \finite\,(S) }{\textbf{H} \;\;\vdash \;\; \finite\,(f)} the user has to write the set corresponding to S \pfun T in the editing area of the Proof Control Window M


*
FIN_FUN2_R
 \frac{\textbf{H} \;\;\vdash\;\;{WD}(S\pfun T) \qquad\textbf{H} \;\;\vdash \;\; f^{-1} \;\in\; S \pfun T \qquad \textbf{H} \;\;\vdash \;\; \finite\,(S) }{\textbf{H} \;\;\vdash \;\; \finite\,(f)} the user has to write the set corresponding to S \pfun T in the editing area of the Proof Control Window M


*
FIN_FUN_IMG_R
 \frac{\textbf{H} \;\;\vdash\;\;{WD}(S\pfun T) \qquad\textbf{H} \;\;\vdash \;\; f \;\in\; S \pfun T \qquad \textbf{H} \;\;\vdash \;\; \finite\,(s) }{\textbf{H} \;\;\vdash \;\; \finite\,(f[s])} the user has to write the set corresponding to S \pfun T in the editing area of the Proof Control Window M


*
FIN_FUN_RAN_R
 \frac{\textbf{H} \;\;\vdash\;\;{WD}(S\pfun T) \qquad\textbf{H} \;\;\vdash \;\; f \;\in\; S \pfun T \qquad \textbf{H} \;\;\vdash \;\; \finite\,(S) }{\textbf{H} \;\;\vdash \;\; \finite\,(\ran(f))} the user has to write the set corresponding to S \pfun T in the editing area of the Proof Control Window M


*
FIN_FUN_DOM_R
 \frac{\textbf{H} \;\;\vdash\;\;{WD}(S\pfun T) \qquad\textbf{H} \;\;\vdash \;\; f^{-1} \;\in\; S \pfun T \qquad \textbf{H} \;\;\vdash \;\; \finite\,(S) }{\textbf{H} \;\;\vdash \;\; \finite\,(\dom(f))} the user has to write the set corresponding to S \pfun T in the editing area of the Proof Control Window M


*
LOWER_BOUND_L
 \frac{\textbf{H} \;\;\vdash \;\; \finite(S) }{\textbf{H} \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; n \leq x)} S must not contain any bound variable M


*
LOWER_BOUND_R
 \frac{\textbf{H} \;\;\vdash \;\; \finite(S) }{\textbf{H} \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; x \geq n)} S must not contain any bound variable M


*
UPPER_BOUND_L
 \frac{\textbf{H} \;\;\vdash \;\; \finite(S) }{\textbf{H} \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; n \geq x)} S must not contain any bound variable M


*
UPPER_BOUND_R
 \frac{\textbf{H} \;\;\vdash \;\; \finite(S) }{\textbf{H} \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; x \leq n)} S must not contain any bound variable M


*
FIN_LT_0
 \frac{\textbf{H} \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; n \leq x) \qquad \textbf{H} \;\;\vdash \;\; S \subseteq \intg \setminus \natn }{\textbf{H} \;\;\vdash \;\; \finite(S)} M


*
FIN_GE_0
 \frac{\textbf{H} \;\;\vdash \;\; \exists n\,\qdot\, (\forall x \,\qdot\, x \in S \;\limp\; x \leq n) \qquad \textbf{H} \;\;\vdash \;\; S \subseteq \nat }{\textbf{H} \;\;\vdash \;\; \finite(S)} M


CARD_INTERV
 \frac{\textbf{H},\, a \leq b \;\;\vdash \;\; \textbf{Q}(b-a+1) \qquad \textbf{H},\, b < a \;\;\vdash \;\; \textbf{Q}(0) }{\textbf{H} \;\;\vdash\;\; \textbf{Q}(\card\,(a\upto b))} where \mathbf{Q} is WD strict M


CARD_EMPTY_INTERV
 \frac{\textbf{H},\, a \leq b,\,\textbf{P}(b-a+1) \;\;\vdash \;\; \textbf{Q} \qquad \textbf{H},\, b < a ,\, \textbf{P}(0)\;\;\vdash \;\; \textbf{Q} }{\textbf{H},\,\textbf{P}(\card\,(a\upto b)) \;\;\vdash\;\; \textbf{Q}} where \mathbf{P} is WD strict M


*
DERIV_LE_CARD
 \frac{\textbf{H} \;\;\vdash\;\; S \subseteq T}{\textbf{H} \;\;\vdash\;\; \card(S) \leq \card(T)} S and T bear the same type M


*
DERIV_GE_CARD
 \frac{\textbf{H} \;\;\vdash\;\; T \subseteq S}{\textbf{H} \;\;\vdash\;\; \card(S) \geq \card(T)} S and T bear the same type M


*
DERIV_LT_CARD
 \frac{\textbf{H} \;\;\vdash\;\; S \subset T}{\textbf{H} \;\;\vdash\;\; \card(S) < \card(T)} S and T bear the same type M


*
DERIV_GT_CARD
 \frac{\textbf{H} \;\;\vdash\;\; T \subset S}{\textbf{H} \;\;\vdash\;\; \card(S) > \card(T)} S and T bear the same type M


*
DERIV_EQUAL_CARD
 \frac{\textbf{H} \;\;\vdash\;\; S = T}{\textbf{H} \;\;\vdash\;\; \card(S) = \card(T)} S and T bear the same type M


SIMP_CARD_SETMINUS_L
 \frac{\textbf{H},\, \textbf{P}(\card (S \setminus T)) \;\;\vdash\;\; \finite(S) \qquad \textbf{H},\, \textbf{P}(\card(S) - \card(S\binter T)) \;\;\vdash\;\; \textbf{G}}{\textbf{H},\, \textbf{P}(\card (S \setminus T)) \;\;\vdash\;\; \textbf{G}} M
SIMP_CARD_SETMINUS_R
 \frac{\textbf{H} \;\;\vdash\;\; \finite(S) \qquad \textbf{H} \;\;\vdash\;\; \textbf{P}(\card(S) - \card(S\binter T))}{\textbf{H} \;\;\vdash\;\; \textbf{P}(\card (S \setminus T))} M


SIMP_CARD_CPROD_L
 \frac{\textbf{H},\, \textbf{P}(\card (S \cprod T)) \;\;\vdash\;\; \finite(S) \qquad \textbf{H},\, \textbf{P}(\card (S \cprod T)) \;\;\vdash\;\; \finite(T) \qquad \textbf{H},\, \textbf{P}(\card(S) * \card(T)) \;\;\vdash\;\; \textbf{G}}{\textbf{H},\, \textbf{P}(\card (S \cprod T)) \;\;\vdash\;\; \textbf{G}} M
SIMP_CARD_CPROD_R
 \frac{\textbf{H} \;\;\vdash\;\; \finite(S) \qquad \textbf{H} \;\;\vdash\;\; \finite(T) \qquad \textbf{H} \;\;\vdash\;\; \textbf{P}(\card(S) * \card(T))}{\textbf{H} \;\;\vdash\;\; \textbf{P}(\card (S \cprod T))} M


*
FORALL_INST
 \frac{\textbf{H} \;\;\vdash \;\; {WD}(E) \qquad \textbf{H} , [x \bcmeq E]\textbf{P} \;\;\vdash \;\; \textbf{G}}{\textbf{H}, \forall x \qdot \textbf{P} \;\;\vdash\;\; \textbf{G}} x is instantiated with E M


*
FORALL_INST_MP
 \frac{\textbf{H} \;\;\vdash \;\; {WD}(E) \qquad \textbf{H}, {WD}(E) \;\;\vdash \;\; [x \bcmeq E]\textbf{P} \qquad \textbf{H}, {WD}(E), [x \bcmeq E]\textbf{Q} \;\;\vdash \;\; \textbf{G}}{\textbf{H}, \forall x \qdot \textbf{P} \limp \textbf{Q} \;\;\vdash\;\; \textbf{G}} x is instantiated with E and a Modus Ponens is applied M


*
FORALL_INST_MT
 \frac{\textbf{H} \;\;\vdash \;\; {WD}(E) \qquad \textbf{H}, {WD}(E) \;\;\vdash \;\; [x \bcmeq E]\lnot\textbf{Q} \qquad \textbf{H}, {WD}(E), [x \bcmeq E]\lnot\textbf{P} \;\;\vdash \;\; \textbf{G}}{\textbf{H}, \forall x \qdot \textbf{P} \limp \textbf{Q} \;\;\vdash\;\; \textbf{G}} x is instantiated with E and a Modus Tollens is applied M


*
CUT
 \frac{\textbf{H} \;\;\vdash \;\; {WD}(\textbf{P}) \qquad \textbf{H}, {WD}(\textbf{P}) \;\;\vdash \;\; \textbf{\textbf{P}} \qquad \textbf{H}, {WD}(\textbf{P}), \textbf{P} \;\;\vdash \;\; \textbf{G}}{\textbf{H} \;\;\vdash\;\; \textbf{G}} hypothesis \textbf{P} is added M


*
EXISTS_INST
 \frac{\textbf{H} \;\;\vdash \;\; {WD}(E) \qquad \textbf{H} \;\;\vdash \;\; \textbf{P}(E)}{\textbf{H} \;\;\vdash\;\; \exists x \qdot \textbf{P}(x)} x is instantiated with E M


*
DISTINCT_CASE
 \frac{\textbf{H} \;\;\vdash \;\; {WD}(\textbf{P}) \qquad \textbf{H}, {WD}(\textbf{P}), \textbf{P} \;\;\vdash \;\; \textbf{\textbf{G}} \qquad \textbf{H}, {WD}(\textbf{P}), \lnot \textbf{P} \;\;\vdash \;\; \textbf{G}}{\textbf{H} \;\;\vdash\;\; \textbf{G}} case distinction on predicate \textbf{P} M


*
ONE_POINT_L
 \frac{\textbf{H} \;\;\vdash \;\; {WD}(E) \qquad \textbf{H}, \forall x, \ldots, \ldots,z \qdot [y \bcmeq E]\textbf{P} \land \ldots \land \ldots \land [y \bcmeq E]\textbf{Q} \limp [y \bcmeq E]\textbf{R} \;\;\vdash \;\; \textbf{G}}{ \textbf{H}, \forall x, \ldots, y, \ldots, z \qdot \textbf{P} \land \ldots \land y = E \land \ldots \land \textbf{Q} \limp \textbf{R} \;\;\vdash\;\; \textbf{G}} The rule can be applied with \forall as well as with \exists A


*
ONE_POINT_R
 \frac{\textbf{H} \;\;\vdash \;\; {WD}(E) \qquad \textbf{H} \;\;\vdash \;\; \forall x, \ldots, \ldots,z \qdot [y \bcmeq E]\textbf{P} \land \ldots \land \ldots \land [y \bcmeq E]\textbf{Q} \limp [y \bcmeq E]\textbf{R} }{ \textbf{H} \;\;\vdash\;\; \forall x, \ldots, y, \ldots, z \qdot \textbf{P} \land \ldots \land y = E \land \ldots \land \textbf{Q} \limp \textbf{R} } The rule can be applied with \forall as well as with \exists A


*
SIM_OV_REL
 \frac{\textbf{H}\vdash x\in A\qquad \textbf{H}\vdash y\in B}{\textbf{H}, f\in A\;op\; B\vdash f\ovl\left\{x\mapsto y\right\}\in A\rel B} where \mathit{op} is one of \rel, \trel, \srel, \strel, \pfun, \tfun, \pinj, \tinj, \psur, \tsur, \tbij A


*
SIM_OV_TREL
 \frac{\textbf{H}\vdash x\in A\qquad \textbf{H}\vdash y\in B}{\textbf{H}, f\in A\;op\; B\vdash f\ovl\left\{x\mapsto y\right\}\in A\trel B} where \mathit{op} is one of \trel, \strel, \tfun,\tinj, \tsur, \tbij A


*
SIM_OV_PFUN
 \frac{\textbf{H}\vdash x\in A\qquad \textbf{H}\vdash y\in B}{\textbf{H}, f\in A\;op\; B\vdash f\ovl\left\{x\mapsto y\right\}\in A\pfun B} where \mathit{op} is one of \pfun, \tfun, \pinj, \tinj, \psur, \tsur, \tbij A


*
SIM_OV_TFUN
 \frac{\textbf{H}\vdash x\in A\qquad \textbf{H}\vdash y\in B}{\textbf{H}, f\in A\;op\; B\vdash f\ovl\left\{x\mapsto y\right\}\in A\tfun B} where \mathit{op} is one of \tfun, \tinj, \tsur, \tbij A


Those following rules have been implemented in the reasoner GeneralizedModusPonens.

  Name Rule Side Condition A/M
*
GENMP_HYP_HYP
 \frac{P,\varphi(\btrue) \vdash G}{P,\varphi(P) \vdash G} A
*
GENMP_NOT_HYP_HYP
 \frac{\lnot P,\varphi(\bfalse) \vdash G}{\lnot P,\varphi(P) \vdash G} A
*
GENMP_HYP_GOAL
 \frac{P \vdash \varphi(\btrue)}{P \vdash \varphi(P)} A
*
GENMP_NOT_HYP_GOAL
 \frac{\lnot P \vdash \varphi(\bfalse)}{\lnot P \vdash \varphi(P)} A
*
GENMP_GOAL_HYP
 \frac{H,\varphi(\bfalse)\vdash G}{H,\varphi(G)\vdash G} A
*
GENMP_NOT_GOAL_HYP
 \frac{H,\varphi(\btrue)\vdash\neg G}{H,\varphi(G)\vdash\neg G} A
*
GENMP_OR_GOAL_HYP
 \frac{H,\varphi(\bfalse)\vdash G_1\lor\cdots\lor G_i\lor\cdots\lor G_n}{H,\varphi(G_i)\vdash G_1\lor\cdots\lor G_i\lor\cdots\lor G_n} A
*
GENMP_OR_NOT_GOAL_HYP
 \frac{H,\varphi(\btrue)\vdash G_1\lor\cdots\lor\neg G_i\lor\cdots\lor G_n}{H,\varphi(G_i)\vdash G_1\lor\cdots\lor\neg G_i\lor\cdots\lor G_n} A


Thos following rules have been implemented in the MembershipGoal reasoner.

  Name Rule Side Condition A/M


*
SUBSET_SUBSETEQ
 A\subset B\vdash A\subseteq B A
*
DOM_SUBSET
 A\subseteq B\vdash \dom(A)\subseteq\dom(B) A
*
RAN_SUBSET
 A\subseteq B\vdash \ran(A)\subseteq\ran(B) A
*
EQUAL_SUBSETEQ_LR
 A=B\vdash A\subseteq B A
*
EQUAL_SUBSETEQ_RL
 A=B\vdash B\subseteq A A
*
IN_DOM_CPROD
 x\in\dom(A\cprod B)\vdash x\in A A
*
IN_RAN_CPROD
 y\in\ran(A\cprod B)\vdash y\in B A
*
IN_DOM_REL
 x\mapsto y\in f\vdash x\in\dom(f) A
*
IN_RAN_REL
 x\mapsto y\in f\vdash y\in\ran(f) A
*
SETENUM_SUBSET
 \left\{a,\cdots,x,\cdots, z\right\}\subseteq A\vdash x\in A A
*
OVR_RIGHT_SUBSET
 f\ovl\cdots\ovl g\ovl\cdots\ovl h\subseteq A\vdash g\ovl\cdots\ovl h\subseteq A A
*
RELSET_SUBSET_CPROD
 f\in A\;op\;B\vdash f\subseteq A\cprod B where \mathit{op} is one of \rel, \trel, \srel, \strel, \pfun, \tfun, \pinj, \tinj, \psur, \tsur, \tbij A
*
DERIV_IN_SUBSET
 x\in A,\;\; A\subseteq B\vdash x\in B A


\textbf{P} \textbf{P}^{\dagger} \textbf{nP}^{\dagger} GenMP A/M
a = b b = a \lnot b = a
a < b \lnot a \ge b, \lnot b \le a, b > a a \ge b, b \le a, \lnot b > a
a > b \lnot a \le b, \lnot b \ge a, b < a a \le b, b \ge a, \lnot b < a
a \le b \lnot a > b, \lnot b < a, b \ge a a > b, b < a, \lnot b \ge a
a \ge b \lnot a < b, \lnot b > a, b \le a a < b, b > a, \lnot b \le a
a \subseteq b
a \subset b
\lnot a = b b = a
\lnot a < b \lnot a > b, \lnot b < a, b \ge a
\lnot a > b \lnot a \le b, \lnot b \ge a, b < a
\lnot a \le b \lnot a > b, \lnot b < a, b \ge a
\lnot a \ge b \lnot a < b, \lnot b > a, b \le a
\lnot a \subseteq b
\lnot a \subset b
\textbf{P} \textbf{P} \neg\, \textbf{P}


See also Extension Proof Rules#Inference Rules.

Retrieved from ‘https://wiki.event-b.org/index.php?title=Inference_Rules&oldid=5763’